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The formatted source code for this file is here.
And a raw version here.
Previous work by Youngser Park can be found here.

We now have the following data sets:

1 Level 0

dat <- fs

1.1 Heat maps (Lv 0):

## Formatting data for heatmap
aggp <- apply(dat, 2, mean)
aggp <- t(cbind(aggp, aggp))[, ford]

The following are heatmaps generated from clustering via K-means++ (at level 1)

heatmap.2(as.matrix(aggp),dendrogram='none',Colv=NA,trace="none", 
          col=mycol,colCol=ccol[ford],cexRow=0.8, keysize=1.25,symkey=FALSE,
          symbreaks=FALSE,scale="none", srtCol=90,main="Heatmap of `fs` data.", 
          labRow = "") 
Figure 1: Heatmap of the cluster means vs channels. Rows and columns are rearranged according to synapse type.

Percentage of data within cluster is presented on the right side of the heatmap.

1.2 Jittered scatter plot: Lv 0

set.seed(1024)
s2 <- sample(dim(dat)[1], 1e4)
ggJdat <- data.table(cbind(stack(dat[s2]),L[s2]))
ggJdat$ind <- as.factor(ggJdat$ind)
levels(ggJdat$ind) <- names(dat)[ford]

ggJ0 <- 
  ggplot(data = ggJdat, aes(x = ind, y = values)) +
  geom_point(alpha=0.75) + 
  geom_jitter(width = 1) + 
  geom_boxplot(alpha =0.35, outlier.color = 'NA') + 
  theme(axis.title.x = element_blank()) + 
  theme(axis.text.x = element_text(color = ccol[ford], 
                                   angle=45,
                                   vjust = 0.5))
print(ggJ0)
Figure 2: Scatter Plot Level 0

The above scatter plot is a random sample of the data points.

1.3 Correlations: Lv 0

cmatfs <- cor(fs)[ford, ford]
corrplot(cmatfs,method="color",tl.col=ccol[ford], tl.cex=1)
Figure 3: Correlation on untransformed F0 data, reordered by synapse type.

1.3.1 PCA of the within cluster correlation matrices at level 0.

pcaL0 <- prcomp(cmatfs, center=TRUE, scale=TRUE)

1.3.1.1 PCA Lv0

pca0 <- data.frame(pcaL0$x)
rgl::plot3d(pca0[,1],pca0[,2],pca0[,3],type='s',col=ccol3[ford], size=1,
            xlab = "PC1", ylab = "PC2", zlab = "PC3")
rgl::rgl.texts(pca0[,1],pca0[,2],pca0[,3],abbreviate(rownames(pca0)), col=ccol3[ford], adj=c(0,2))
title3d(main = "Lv 0")
subid <- currentSubscene3d()
rglwidget(elementId="rgl-pca0",width=720,height=720)

1.3.2 LDA on Lv 0

Using LDA with re-substitution to create a voronoi diagram for Lv 0.

tr <- factor(exType, ordered = FALSE)
lda.fit0 <- lda(tr ~ ., data = pca0[, 1:10])
lda.pred0 <- predict(lda.fit0)

titlesvor <- paste("LDA decision boundaries for", paste0("F", 0:5))
voronoidf <- data.frame(lda.fit0$means)
#voronoidf <- data.frame(x=lda.fit$means[,1],y=lda.fit$means[,2])

#This creates the voronoi line segments

plot(pca0[,1:2], col=ccol3[ford], pch=20, cex=1.5, xlim =
     c(min(pca0[,1:2]) -3, 5))
text(pca0[,1:2], labels=rownames(pca0), 
     pos=ifelse(exType %in% c('ex', 'in'), 2, 4), 
     col=ccol3[ford], cex=1.2)

deldir(x = voronoidf[,1],y = voronoidf[,2], 
       rw = c(-15,15,-15,15), 
       plotit=TRUE, add=TRUE, wl='te')
text(voronoidf[,1:2], labels=rownames(voronoidf), cex=1.25, pos=2)
Figure 5: LDA for 3 classes on Lv 0

1.4 Mean-difference Explorations

Staring from the correlation matrix cmatfs we compute the class means; \(v_1 = column\_mean(\)Excitatory\()\), \(v_2 = column\_mean(\)Inhibatory\()\), \(v_3 = column\_mean(\)Other\()\).

We then compute \(r_1 = v_1 - v_2\) and \(r_2 = v_1 - v_3\).

We then multiply \([cmatfs] \cdot [r_1 |\, r_2]\). The transformed points are then plotted below.

exMat <- cmatfs[exType == 'ex',] -> g1
inMat <- cmatfs[exType == 'in',] -> g2
otMat <- cmatfs[exType == 'other',] -> g3

mEx <- apply(exMat, 2, mean) -> v1   
mIn <- apply(inMat, 2, mean) -> v2 
mOt <- apply(otMat, 2, mean) -> v3 

mXI <- mEx - mIn
mXO <- mEx - mOt

rotM <- data.frame(XI = mXI, XO = mXO) -> r1r2
colnames(r1r2) <- c("r1", "r2")

mdm2 <- data.frame(cmatfs %*% as.matrix(rotM))
mdm1 <- data.frame(XI = mdm2[,1], row.names=rownames(mdm2))
newFord <- order(mdm2$XI)

1.4.1 Within Group Heatmaps

heatmap.2(as.matrix(g1[, newFord]),dendrogram='none',
          Colv=NA,trace="none", 
          col=mycol,colCol=ccol3[ford][newFord],
          cexRow=0.8, keysize=1.25,
          symkey=FALSE,symbreaks=FALSE,
          scale="none", srtCol=90,
          main="Excitatory") 
Figure 6: Heatmaps of groups Ex, In, Ot

heatmap.2(as.matrix(g2[, newFord]),dendrogram='none',
          Colv=NA,trace="none", 
          col=mycol,colCol=ccol3[ford][newFord],
          cexRow=0.8, keysize=1.25,
          symkey=FALSE,symbreaks=FALSE,
          scale="none", srtCol=90,
          main="Inhibatory") 
Figure 6: Heatmaps of groups Ex, In, Ot

heatmap.2(as.matrix(g3[, newFord]),dendrogram='none',
          Colv=NA,trace="none", 
          col=mycol,colCol=ccol3[ford][newFord],
          cexRow=0.8, keysize=1.25,
          symkey=FALSE,symbreaks=FALSE,
          scale="none", srtCol=90,
          main="Other") 
Figure 6: Heatmaps of groups Ex, In, Ot

1.4.2 Group Mean Heatmaps

tmp <- t(cbind(v1, v1))[, newFord]
heatmap.2(as.matrix(tmp),dendrogram='none',
          Colv=NA,trace="none", key= FALSE,
          col=mycol,colCol=ccol3[ford][newFord],
          cexRow=0.8, keysize=1.25,
          symkey=FALSE,symbreaks=FALSE,
          scale="none", srtCol=90, labRow='',
          main="v1 = Mean Excitatory") 
Figure 7: Heatmaps of v1, v2, v3

tmp <- t(cbind(v2, v2))[, newFord]
heatmap.2(as.matrix(tmp),dendrogram='none',
          Colv=NA,trace="none", key = FALSE,
          col=mycol,colCol=ccol3[ford][newFord],
          cexRow=0.8, keysize=1.25,
          symkey=FALSE,symbreaks=FALSE,
          scale="none", srtCol=90, labRow='',
          main="v2 = Mean Inhibatory") 
Figure 7: Heatmaps of v1, v2, v3

tmp <- t(cbind(v3, v3))[, newFord]
heatmap.2(as.matrix(tmp),dendrogram='none',
          Colv=NA,trace="none", key = FALSE,
          col=mycol,colCol=ccol3[ford][newFord],
          cexRow=0.8, keysize=1.25,
          symkey=FALSE,symbreaks=FALSE,
          scale="none", srtCol=90, labRow='',
          main="v3 = Mean Other") 
Figure 7: Heatmaps of v1, v2, v3

rm(tmp)

1.4.3 Mean Difference Vectors Heatmap

heatmap.2(as.matrix(t(r1r2)[, newFord]),dendrogram='none',
          Colv=NA,trace="none", 
          col=mycol,colCol=ccol3[ford][newFord],
          cexRow=0.8, keysize=1.25,
          symkey=FALSE,symbreaks=FALSE,
          scale="none", srtCol=90,
          main="r1r2") 
Figure 8: Heatmap of r1 and r2.

1.4.4 Mean-difference Marker Projections

plot(mdm2, col = ccol3[ford], pch = 20, cex=1)
abline(h = 0, v = 0)
text(mdm2, labels=rownames(mdm2), col=ccol3[ford], cex = 0.75, pos=1)


lm.fit <- lm(mdm2[,2] ~ mdm2[,1])
rL <- list(a = lm.fit$coefficients[1],
          b = lm.fit$coefficients[2])
L1 <- function(x){ rL$a + rL$b * x }

summary(lm.fit)

abline(rL$a,rL$b, col = 'darkorange', lty=2)
text(-0.1,L1(0.1) , labels=expression(L[1]), col='darkorange', cex = 1) 

legend(list(x= -1.5,y=1.6), legend = c('regression line'), col = 'darkorange', lty=2)
title("Mean-difference projections of markers")
Figure 9: Mean-difference projections 2d

# 
# Call:
# lm(formula = mdm2[, 2] ~ mdm2[, 1])
# 
# Residuals:
#      Min       1Q   Median       3Q      Max 
# -0.39388 -0.23497  0.01289  0.20515  0.39076 
# 
# Coefficients:
#             Estimate Std. Error t value Pr(>|t|)    
# (Intercept)  0.14481    0.05626   2.574   0.0173 *  
# mdm2[, 1]    0.72839    0.05723  12.726 1.28e-11 ***
# ---
# Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# 
# Residual standard error: 0.2636 on 22 degrees of freedom
# Multiple R-squared:  0.8804,  Adjusted R-squared:  0.875 
# F-statistic:   162 on 1 and 22 DF,  p-value: 1.276e-11

The following is the projection onto \(r_1\).

set.seed(2^8)
print(ggplot(mdm1, aes(x = XI, y = 0, label=rownames(mdm1))) +
  geom_point(color = ccol3[ford]) + 
  geom_text(color = ccol3[ford], check_overlap=FALSE,
            position='jitter'))
Figure 10: Mean-difference projections 1d

We will now project the individual synapse points onto the
line \(L_1\) by first projecting into the mean-difference space then projecting onto \(L_1\) followed by a rotation into \(\mathbb{R}\).

mfs <- as.matrix(fs)
r <- as.matrix(mdm1)
projR1 <- mfs %*% r

plot(density(projR1))

if(FALSE){
require(ggvis)
data.frame(projR1) %>% ggvis(x = ~XI) %>%
    layer_densities(
      adjust = input_slider(.1, 2, value = 1, step = .1, label = "Bandwidth adjustment"),
      kernel = input_select(
        c("Gaussian" = "gaussian",
          "Epanechnikov" = "epanechnikov",
          "Rectangular" = "rectangular",
          "Triangular" = "triangular",
          "Biweight" = "biweight",
          "Cosine" = "cosine",
          "Optcosine" = "optcosine"),
        label = "Kernel")
    )
}
svdfs <- svd(as.matrix(fs)[,newFord])

rightSV <- data.frame(t(svdfs$v[,1:2]))
colnames(rightSV) <- rownames(mdm2[order(mdm2$XI),])

heatmap.2(as.matrix(rightSV),dendrogram='row',Colv=NA,trace="none", col=mycol,colCol=ccol3[ford][newFord],cexRow=0.8, keysize=1.25,symkey=FALSE,symbreaks=FALSE,scale="none", srtCol=90,main="r. singular vectors of 'fs' data") 
Figure 11: Heatmap of 1:2 right singular vectors of ‘fs’ data ordered by projection on L_1.

2 Level 1: K-means++ for \(K=2\).

We run a Hierachical K-means++ for \(K=2\) on the fs data with 4 levels.

set.seed(2^13)
L <- bhkmpp(dat,blevels=4)

2.1 Heat maps (Lv 1):

## Formatting data for heatmap
aggp <- aggregate(dat,by=list(lab=L[[1]]),FUN=mean)
aggp <- as.matrix(aggp[,-1])[, ford]
rownames(aggp) <- clusterFraction(L[[1]])

The following are heatmaps generated from clustering via K-means++ (at level 1)

heatmap.2(as.matrix(aggp),dendrogram='row',Colv=NA,trace="none", col=mycol,colCol=ccol[ford],cexRow=0.8, keysize=1.25,symkey=FALSE,symbreaks=FALSE,scale="none", srtCol=90,main="Heatmap of `fs` data.") 
Figure 12: Heatmap of the cluster means vs channels. Rows and columns are rearranged according to synapse type.

Percentage of data within cluster is presented on the right side of the heatmap.

2.2 Jittered scatter plot: Lv 1

ggCol <- brewer.pal(4,"Set1")[order(table(L[[1]]))]

cf1 <- data.frame(cf = clusterFraction(L[[1]]))

ggJ1 <- 
  ggplot(data = ggJdat, aes(x = ind, y = values, 
                         color = as.factor(lv1))) +
  scale_color_manual(values=ggCol, name="Cluster") + 
  geom_point(alpha=0.25, position=position_jitterdodge()) + 
  geom_boxplot(alpha =0.35, outlier.color = 'NA') + 
  annotate("text", x = levels(ggJdat$ind)[c(2,20)], y = 1.15*max(ggJdat$values), 
           label= cf1[1:2,]) + 
  theme(axis.title.x = element_blank()) + 
  theme(axis.text.x = element_text(color = ccol[ford], 
                                   angle=45,
                                   vjust = 0.5))
print(ggJ1)
Figure 13: Scatter Plot Level 1

2.3 Within cluster correlations (Lv 1)

corkp1 <- cor(dat[L[[1]] == 1,])[ford, ford]
corkp2 <- cor(dat[L[[1]] == 2,])[ford, ford]
difCor12 <- (corkp1 - corkp2)

layout(matrix(c(1,2,3,3), 2, 2, byrow=TRUE))
corrplot(corkp1,method="color",tl.col=ccol[ford], tl.cex=0.8, mar=c(0,0,3,0))
title("Cluster 1")
corrplot(corkp2,method="color",tl.col=ccol[ford], tl.cex=0.8, mar=c(0,0,3,0))
title("Cluster 2")
corrplot(difCor12,is.corr=FALSE,method="color",
         tl.col=ccol[ford], tl.cex=0.8, 
         mar=c(0,0,3,0),
         col=colorRampPalette(c("#998ec3","white","darkorange"))(50))
title("Difference(1,2)")
Figure 14: Within cluster correlations, clock-wise from top left, Cluster 1, Cluster 2, difference C1 - C2

Notice that the non-synaptic markers change very little between clusters. Also note that the correlations between (gad, VGAT, PV, Gephyr) and VGlut1 at both times change significantly between clusters.

Next we compute PCA for the within cluster correlation matrices and embed in 3d.

2.3.1 PCA of the within cluster correlation matrices at level 1.

pcaL <- lapply(list(corkp1, corkp2), prcomp, center=TRUE, scale=TRUE)
elB <- lapply(pcaL, function(x) {getElbows(x$sdev, plot=FALSE)})

2.3.1.1 PCA C1 Lv1

pca <- pcaL[[1]]$x
rgl::plot3d(pca[,1],pca[,2],pca[,3],type='s',col=ccol3[ford], size=1,
            xlab = "PC1", ylab = "PC2", zlab = "PC3")
rgl::rgl.texts(pca[,1],pca[,2],pca[,3],abbreviate(rownames(pca)), col=ccol3[ford], adj=c(0,2))
title3d(main = "Cluster 1: Lv 1")
subid <- currentSubscene3d()
rglwidget(elementId="rgl-pca1",width=720,height=720)

2.3.1.2 PCA C1 Lv1

pca <- pcaL[[2]]$x
rgl::plot3d(pca[,1],pca[,2],pca[,3],type='s',col=ccol3[ford], size=1,
            xlab = "PC1", ylab = "PC2", zlab = "PC3")
rgl::rgl.texts(pca[,1],pca[,2],pca[,3],abbreviate(rownames(pca)), col=ccol3[ford], adj=c(0,2))
title3d(main = "Cluster 2: Lv 1")
subid <- currentSubscene3d()
rglwidget(elementId="rgl-pca2",width=720,height=720)

2.3.2 LDA on Lv1

lda.fit <- 
  lapply(list(pcaL[[1]]$x[,1:elB[[1]][2]],
              pcaL[[2]]$x[,1:elB[[2]][2]]),
                function(y) {
                  lda(tr ~ ., data = as.data.frame(y))
                })

titlesvor <- paste("LDA decision boundaries for", paste0("C", 1:2))
voronoidf <- lapply(lapply(lda.fit, '[[', 3), data.frame)

#This creates the voronoi line segments

par(mfrow = c(1,2))
for(i in 1:length(pcaL)){
  plot(pcaL[[i]]$x[,1:2], col=ccol3[ford], pch=20, cex=1.5)
  title(titlesvor[i])
  text(pcaL[[i]]$x[,1:2], labels=rownames(pcaL[[i]]$x), 
       pos=ifelse(pcaL[[i]]$x[,1]<max(pcaL[[i]]$x[,1] -0.5),4,2), 
       col=ccol3[ford], cex=1.2)

  deldir(x = voronoidf[[i]][,1],y = voronoidf[[i]][,2], rw = c(-15,15,-15,15), 
       plotit=TRUE, add=TRUE, wl='te')
  text(voronoidf[[i]], labels=rownames(voronoidf[[i]]), cex=1.5, pos=1)
}
Figure 18: Voronoi diagrams on class means from LDA on PCA of untransformed correlation matrices

2.4 Clusters and Spatial Location (Lv 1)

Using the location data and the results of K-means++ we show a 3d scatter plot colored accoding to cluster.

set.seed(2^12)
s1 <- sample(dim(loc)[1],5e4)

locs1 <- loc[s1,]
locs1$cluster <- L[[1]][s1]

plot3d(locs1$V1,locs1$V2,locs1$V3,
       col=brewer.pal(4,"Set1")[order(table(L[[1]]))][locs1$cluster],
       alpha=0.75,
       xlab='x', 
       ylab='y', 
       zlab='z')

subid <- currentSubscene3d()
rglwidget(elementId="plot3dLocations", height=720, width=720)

3 Level 2: K-means++ for \(K=2\).

3.1 Heat maps (Lv 2):

## Formatting data for heatmap
aggp2 <- aggregate(dat,by=list(lab=L[[2]]),FUN=function(x){mean(x)}) 
aggp2 <- as.matrix(aggp2[,-1])[, ford]
rownames(aggp2) <- clusterFraction(L[[2]])

The following are heatmaps generated from clustering via K-means++

heatmap.2(as.matrix(aggp2),dendrogram='row',Colv=NA,trace="none", col=mycol,colCol=ccol[ford],cexRow=0.8, keysize=1.25,symkey=FALSE,symbreaks=FALSE,scale="none", srtCol=90,main="Heatmap of `fs` data.") 
Figure 19: Heatmap of the cluster means vs channels. Rows and columns are rearranged according to synapse type.

Percentage of data within cluster is presented on the right side of the heatmap.

3.2 Jittered scatter plot: Lv 2

ggCol <- brewer.pal(8,"Set1")[order(table(L[[2]]))]
cf2 <- data.frame(cf = clusterFraction(L[[2]]))

ggJ2 <- 
  ggplot(data = ggJdat, aes(x = ind, y = values, 
                         color = as.factor(lv2))) +
  scale_color_manual(values=ggCol, name="Cluster") + 
  geom_point(alpha=0.25, position=position_jitterdodge()) + 
  geom_boxplot(alpha =0.35, outlier.color = 'NA') + 
  annotate("text", x = levels(ggJdat$ind)[c(2,8,14,20)], y = 1.15*max(ggJdat$values), 
           label= cf2[1:4,]) + 
  theme(axis.title.x = element_blank()) + 
  theme(axis.text.x = element_text(color = ccol[ford], 
                                   angle=45,
                                   vjust = 0.5))
print(ggJ2)
Figure 20: Scatter Plot Level 2

The fraction of data points within each cluster are given at the top of the plot window.

3.3 Within cluster correlations (Lv 2)

corLV2 <- lapply(c(1:4),function(x){cor(dat[L[[2]] == x,])[ford, ford]})

difCor1112 <- ((corLV2[[1]] - corLV2[[2]]))
difCor2122 <- ((corLV2[[3]] - corLV2[[4]]))

layout(matrix(c(1,2,3,3,4,5,6,6), 4, 2, byrow=TRUE))
corrplot(corLV2[[1]],method="color",tl.col=ccol[ford], tl.cex=0.8, 
         mar=c(0,0,3,0))
title("Cluster 1")
corrplot(corLV2[[2]],method="color",tl.col=ccol[ford], tl.cex=0.8, 
         mar=c(0,0,3,0))
title("Cluster 2")
corrplot(difCor1112, method="color", tl.col=ccol[ford], 
         tl.cex=0.8,
         mar = c(0,0,3,0),
         cl.lim = c(min(difCor1112,difCor2122),max(difCor1112,difCor2122)),
         col=colorRampPalette(c("#998ec3", 
                                "white",
                                "darkorange"))(100))
title("Difference(1,2)")
corrplot(corLV2[[3]],method="color",tl.col=ccol[ford], tl.cex=0.8, 
         mar=c(0,0,3,0))
title("Cluster 3")
corrplot(corLV2[[4]],method="color",tl.col=ccol[ford], tl.cex=0.8, 
         mar=c(0,0,3,0))
title("Cluster 4")
corrplot(difCor2122, method="color", tl.col=ccol[ford], 
         tl.cex=0.8,
         mar=c(0,0,3,0),
         cl.lim = c(min(difCor1112,difCor2122),max(difCor1112,difCor2122)),
         col=colorRampPalette(c("#998ec3", 
                                "white",
                                "darkorange"))(100))
title("Difference(3,4)")
Figure 21: Within cluster correlations for level 2. (c11, c12, c21, c22) with differences

3.4 PCA of the within cluster correlation matrices at level 2.

pcaL <- lapply(corLV2, prcomp, center=TRUE, scale=TRUE)
elB <- lapply(pcaL, function(x) {getElbows(x$sdev, plot=FALSE)})
pcaLel2 <- mapply(function(x,y){ x$x[,1:y[2]] }, pcaL, elB)

3.5 LDA on Lv 2

lda.fit <- 
  lapply(pcaLel2, 
         function(y) {
                  lda(tr ~ ., data = as.data.frame(y))
                })

titlesvor <- paste("LDA decision boundaries for", paste0("C", 1:4))
voronoidf <- lapply(lapply(lda.fit, '[[', 3), data.frame)

#This creates the voronoi line segments

par(mfrow = c(2,2))
for(i in 1:length(pcaL)){
  plot(pcaL[[i]]$x[,1:2], col=ccol3[ford], pch=20, cex=1.5)
  title(titlesvor[i])
  text(pcaL[[i]]$x[,1:2], labels=rownames(pcaL[[i]]$x), 
       pos=ifelse(pcaL[[i]]$x[,1]<max(pcaL[[i]]$x[,1] -0.5),4,2), 
       col=ccol3[ford], cex=1.2)

  deldir(x = voronoidf[[i]][,1],y = voronoidf[[i]][,2], rw = c(-15,15,-15,15), 
       plotit=TRUE, add=TRUE, wl='te')
  text(voronoidf[[i]], labels=rownames(voronoidf[[i]]), cex=1.5, pos=1)
}
Figure 23: Voronoi diagrams on class means from LDA on PCA of untransformed correlation matrices

3.6 Clusters and Spatial Location (Lv 2)

Using the location data and the results of K-means++ we show a 3d scatter plot colored according to cluster.

set.seed(2^12)
s1 <- sample(dim(loc)[1],5e4)

locs2 <- loc[s1,]
locs2$cluster <- L[[2]][s1]

YlOrBr <- c("#FFFFD4", "#FED98E", "#FE9929", "#D95F0E", "#993404")
col.pal <- colorRampPalette(YlOrBr)

plot3d(locs2$V1,locs2$V2,locs2$V3,
       #col=colorpanel(8,"brown","blue")[order(table(L[[2]]))][locs2$cluster],
       col=col.pal(8)[-seq(1,8,2)][order(table(L[[2]]))][locs2$cluster],
       alpha=0.75,
       xlab='x', 
       ylab='y', 
       zlab='z'
       )

subid <- currentSubscene3d()
rglwidget(elementId="plot3dLocationsLV2", height=720, width=720)

4 Level 3: K-means++ for \(K=2\).

4.1 Heat maps (Lv 3):

## Formatting data for heatmap
aggp3 <- aggregate(dat,by=list(lab=L[[3]]),FUN=function(x){mean(x)})
aggp3 <- as.matrix(aggp3[,-1])[, ford]
rownames(aggp3) <- clusterFraction(L[[3]])

The following are heatmaps generated from clustering via K-means++

heatmap.2(as.matrix(aggp3),dendrogram='row',Colv=NA,trace="none", col=mycol,colCol=ccol[ford],cexRow=0.8, keysize=1.25,symkey=FALSE,symbreaks=FALSE,scale="none", srtCol=90,main="Heatmap of `fs` data.") 
Figure 24: Heatmap of the cluster means vs channels. Rows and columns are rearranged according to synapse type.

Percentage of data within cluster is presented on the right side of the heatmap.

4.2 Jittered scatter plot: Lv 3

ggCol <- brewer.pal(8,"Set1")[order(table(L[[3]]))]
cf3 <- data.frame(cf = clusterFraction(L[[3]]))

ggJ3 <- 
  ggplot(data = ggJdat, aes(x = ind, y = values, 
                         color = as.factor(lv3))) +
  scale_color_manual(values=ggCol, name="Cluster") + 
  geom_point(alpha=0.25, position=position_jitterdodge()) + 
  geom_boxplot(alpha =0.35, outlier.color = 'NA') + 
  annotate("text", x = levels(ggJdat$ind)[seq(2,22,length=8)], y = 1.05*max(ggJdat$values), 
           label= cf3[1:8,]) + 
  #geom_jitter(width=2) + 
  theme(axis.title.x = element_blank()) + 
  theme(axis.text.x = element_text(color = ccol[ford], 
                                   angle=45,
                                   vjust = 0.5))
print(ggJ3)
Figure 25: Scatter Plot Level 3

4.3 Within cluster correlations (Lv 3)

corLV3 <- lapply(c(1:8),function(x){cor(dat[L[[3]] == x,])[ford, ford]})

difCor1 <- (corLV3[[1]] - corLV3[[2]])
difCor2 <- (corLV3[[3]] - corLV3[[4]])
difCor3 <- (corLV3[[5]] - corLV3[[6]])
difCor4 <- (corLV3[[7]] - corLV3[[8]])
M <- max(difCor1, difCor2, difCor3, difCor4)
m <- min(difCor1, difCor2, difCor3, difCor4)

layout(matrix(c(1, 2, 3, 3,
                4, 5, 6, 6, 
                7, 8, 9, 9,
                10, 11, 12, 12), 8,2, byrow=TRUE))

corrplot(corLV3[[1]],method="color",tl.col=ccol[ford], tl.cex=0.8,
         mar=c(0,0,3,0))
title('Cluster 1')
corrplot(corLV3[[2]],method="color",tl.col=ccol[ford], tl.cex=0.8,
         mar=c(0,0,3,0))
title('Cluster 2')
corrplot(difCor1,method="color",tl.col=ccol[ford], tl.cex=0.8,
         cl.lim=c(m,M), 
         mar=c(0,0,3,0),
         col=colorRampPalette(c("#998ec3",
                                "white",
                                "darkorange"))(50))
title('Difference(1,2)')
corrplot(corLV3[[3]],method="color",tl.col=ccol[ford], tl.cex=0.8, 
         mar=c(0,0,3,0))
title('Cluster 3')
corrplot(corLV3[[4]],method="color",tl.col=ccol[ford], tl.cex=0.8, 
         mar=c(0,0,3,0))
title('Cluster 4')
corrplot(difCor2,method="color",tl.col=ccol[ford], tl.cex=0.8, 
         cl.lim= c(m,M),
         mar=c(0,0,3,0),
         col=colorRampPalette(c("#998ec3",
                                "white",
                                "darkorange"))(50))
title('Difference(3,4)')
corrplot(corLV3[[5]],method="color",tl.col=ccol[ford], tl.cex=0.8, 
         mar=c(0,0,3,0))
title('Cluster 5')
corrplot(corLV3[[6]],method="color",tl.col=ccol[ford], tl.cex=0.8, 
         mar=c(0,0,3,0))
title('Cluster 6')
corrplot(difCor3,method="color",tl.col=ccol[ford], tl.cex=0.8,
         cl.lim= c(m,M),
         mar=c(0,0,3,0),
         col=colorRampPalette(c("#998ec3",
                                "white",
                                "darkorange"))(50))
title('Difference(5,6)')
corrplot(corLV3[[7]],method="color",tl.col=ccol[ford], tl.cex=0.8, 
         mar=c(0,0,3,0))
title('Cluster 7')
corrplot(corLV3[[8]],method="color",tl.col=ccol[ford], tl.cex=0.8, 
         mar=c(0,0,3,0))
title('Cluster 8')
corrplot(difCor4,method="color",tl.col=ccol[ford], tl.cex=0.8,
         cl.lim= c(m,M),
         mar=c(0,0,3,0),
         col=colorRampPalette(c("#998ec3",
                                "white",
                                "darkorange"))(50))
title('Difference(7,8)')
Figure 26: Within cluster correlations for level 3. (c111, c112, c121, c122, c211, c212, c221, c222)

4.4 PCA of the within cluster correlation matrices at level 3.

pcaL <- lapply(corLV3, prcomp, center=TRUE, scale=TRUE)
elB <- lapply(pcaL, function(x) {getElbows(x$sdev, plot=FALSE)})
pcaLel3 <- mapply(function(x,y){ x$x[,1:y[2]] }, pcaL, elB)

4.5 LDA on Lv 3

lda.fit <- 
  lapply(pcaLel3, 
         function(y) {
                  lda(tr ~ ., data = as.data.frame(y))
                })

titlesvor <- paste("LDA decision boundaries for", paste0("C", 1:8))
voronoidf <- lapply(lapply(lda.fit, '[[', 3), data.frame)

#This creates the voronoi line segments

par(mfrow = c(4,2))
for(i in 1:length(pcaL)){
  plot(pcaL[[i]]$x[,1:2], col=ccol3[ford], pch=20, cex=1.5)
  title(titlesvor[i])
  text(pcaL[[i]]$x[,1:2], labels=rownames(pcaL[[i]]$x), 
       pos=ifelse(pcaL[[i]]$x[,1]<max(pcaL[[i]]$x[,1] -0.5),4,2), 
       col=ccol3[ford], cex=1.2)

  deldir(x = voronoidf[[i]][,1],y = voronoidf[[i]][,2], rw = c(-15,15,-15,15), 
       plotit=TRUE, add=TRUE, wl='te')
  text(voronoidf[[i]], labels=rownames(voronoidf[[i]]), cex=1.5, pos=1)
}
Figure 27: Voronoi diagrams on class means from LDA on PCA of untransformed correlation matrices

4.6 Clusters and Spatial Location (Lv 3)

Using the location data and the results of K-means++ we show a 3d scatter plot colored according to cluster.

set.seed(2^12)
s1 <- sample(dim(loc)[1],5e4)

locs3 <- loc[s1,]
locs3$cluster <- L[[3]][s1]

plot3d(locs3$V1,locs3$V2,locs3$V3,
       col=col.pal(16)[-seq(1,8,2)][order(table(L[[3]]))][locs3$cluster],
       alpha=0.65,
       xlab='x', 
       ylab='y', 
       zlab='z'
       )

subid <- currentSubscene3d()
rglwidget(elementId="plot3dLocationsLV3", height=720, width=720)